Weak temporal classification

The simplest type of temporal classification is based on associating a single class label with each stream. In the Tech Support domain, the classification is the outcome, happy or angry, after a phone call. In the sign language problem, the classification is of the sign, based on the samples from a single stream. Each of these are weak temporal classification tasks.

Let $\mathit{SS}$ be a set of streams with the same type. Let $\mathit{CL}$ be a set of labels, that describes the set of possible classes.

Define a function

$$\mathit{class}: \ensuremath{\mathit{SS}}\rightarrow \ensuremath{\mathit{CL}}$$

which takes an element of $\mathit{SS}$ and returns an element of $\mathit{CL}$.

The goal is: given a subset of the function $\mathit{class}$ (say $\ensuremath{\mathit{class}}_T$), produce a function $\mathit{class}_P$ which is as similar to $\mathit{class}$ as possible. The exact meaning of ``similar'' is explored in Section 2.4.

Mathematically, this can be viewed as trying to develop a function $\mathit{L}$:

$$L: \ensuremath{\mathit{class}_T}\rightarrow \ensuremath{\mathit{class}_P}$$

such that the symmetric difference between $\mathit{class}_P$ and $\mathit{class}$ is as small as possible. This is the definition of traditional concept learning and is included here for completeness.

Intuitively, our goal is: given a limited example of streams and their classes, in other words, some subset of the function $\mathit{class}$, can we determine the rest of the function?